582 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII 

 which leads to 



(I*). 



(15). 



The case of an infinitely long elliptic cylinder may also be 

 noticed. Putting c = oo in (5), we find 



26 2a n . 



%-+& ft-j+V 7o = ............ (16). 



The energy per unit length of the cylinder is 



Fl-.A^Wfcg^J ............... (17), 



if a 2 = tt6. 



314. If the ellipsoid rotate in relative equilibrium about the 

 axis of z t with angular velocity ?i, the component accelerations 

 of the particle (#, T/, #) are ?& 2 #, ?i a y, 0, so that the dynamical 

 equations reduce to 



1 dp dl 1 dp dl 1 dp dl 



-n z x = --- -*, n*y = --- f~T, = --- f-~r 



p ax ax p ay ay p az dz 



............... (1). 



Hence ^ = \ n? (a* + f) - O + const ................ (2). 



The surfaces of equal pressure are therefore given by 



In order that one of these may coincide with the external 

 surface 



we must have 



(5). 



In the case of an ellipsoid of revolution (a = b), these con 

 ditions reduce to one : 



